Grade 10 core mathematics Measurements and Geometry – Rotation Notes

1. Identify and outline the sub-sub-strands:
   • Properties of rotation
   • Rotation on different planes
   • Rotational symmetry
   • Rotation and congruence
2. Determine properties of rotation in different situations.
3. Rotate an object given the centre and angle of rotation on a plane surface and on the Cartesian plane.
4. Determine the centre and angle of rotation given an object and its image.
5. Determine the order of rotational symmetry of plane figures.
6. Determine the axis and order of rotational symmetry in solids.
7. Deduce congruence from rotation.
8. Appreciate applications of rotation in real-life situations.

What is a rotation?

A rotation is a transformation that turns every point of a figure about a fixed point called the centre of rotation through a specified angle and direction (clockwise or anticlockwise). Rotations are rigid motions — they preserve distances and angles.

Key properties of rotation

• Distance-preserving: lengths are unchanged (isometry).
• Angle-preserving: all angles remain equal.
• The centre of rotation remains fixed.
• Orientation-preserving (unlike reflection).
• Composition of rotations is another rotation (or a translation if special case of two opposite rotations about different centres).

Rotate a figure on a plane surface (construction with ruler & protractor)

1. Mark the centre of rotation O and the angle of rotation θ and direction (clockwise or anticlockwise).
2. For each vertex A of the figure:
   1. Join O to A. Measure the distance OA with a compass or ruler — this distance does not change.
   2. At O, use a protractor to measure angle θ from OA in the chosen direction. Draw a ray making that angle with OA.
   3. From O on this ray mark a point A' at distance OA (use compass). Repeat for every vertex.
3. Join the images A'B'C' … in the same order to obtain the rotated image.

Visual example: rotation by 90° anticlockwise about the origin

Coordinate rule (about origin): For anticlockwise rotation by angle θ, (x', y') = (x cosθ − y sinθ, x sinθ + y cosθ). Example: 90° CCW: (x', y') = (−y, x). So (2,1) → (−1,2).

Rotation on the Cartesian plane (formulas)

About the origin (0,0):

• General θ: (x', y') = (x cosθ − y sinθ, x sinθ + y cosθ).
• Common special angles:
  • 90° CCW: (x', y') = (−y, x).
  • 180°: (x', y') = (−x, −y).
  • 270° CCW (or 90° CW): (x', y') = (y, −x).
• About a point (h,k): translate so (h,k) → origin, apply rotation, then translate back:
  (x', y') = (h + (x−h)cosθ − (y−k)sinθ, k + (x−h)sinθ + (y−k)cosθ).

How to determine centre and angle of rotation from a figure and its image

1. Choose two non-collinear points A and B and their images A' and B'.
2. Construct the perpendicular bisector of segment AA' and the perpendicular bisector of BB'.
3. Their intersection O is the centre of rotation (since O is equidistant from A and A' and from B and B').
4. The angle of rotation θ is the angle between OA and OA' (measure direction to decide clockwise or anticlockwise).

Remark: If all perpendicular bisectors do not meet at a single point, the transformation is not a pure rotation.

Rotational symmetry

Definition: A figure has rotational symmetry if it can be rotated about a point (or an axis, for solids) by an angle less than 360° and look exactly the same. The order of rotational symmetry is the number of times the figure maps onto itself during a full 360° rotation.

• Plane figures:
  • Regular n-gon has order n (e.g., equilateral triangle order 3, square order 4, regular pentagon order 5).
  • A rectangle (not square) has order 2 (rotations by 180° and 360°).
  • A scalene triangle typically has order 1 (only the identity).
• Solids (axis and order):
  • Cube: 3 axes through opposite faces (4-fold), 4 axes through opposite vertices (3-fold). Many axes — consider the main ones: face-centred axes give order 4.
  • Cylinder: infinite order about its central axis (continuous symmetry).
  • Regular tetrahedron: has 3-fold rotations through axes joining a vertex to the opposite face centre (order 3).

Rotation and congruence

If one figure can be obtained from another by a rotation (possibly together with translations or reflections depending on context), the two figures are congruent: corresponding sides and angles are equal. In particular, a rotation alone is a congruence mapping because it preserves distances and angles.

Example: If triangle ABC is rotated about O to give A'B'C', then AB = A'B', BC = B'C', CA = C'A' and corresponding angles are equal; hence ΔABC ≅ ΔA'B'C'.

Real-life applications (Kenyan context)

• Bicycle and motorbike wheels — rotation about an axle; study of rotational symmetry of spokes.
• Grain mill (hand or mechanised) and water wheel — rotational motion and axis.
• Textile and bead patterns (kitenge, Maasai bead designs) show rotational symmetry and repeated motifs.
• Wind-turbine or water-pump blades — axis of rotation and symmetry for balance.
• Design tasks: creating logos, shields or rosette patterns that have a given order of rotational symmetry.

Suggested learning experiences (classroom & practical)

• Hands-on construction: Give pupils tracing paper, a protractor and compass. Rotate simple shapes (triangles, letters, motifs) about a chosen centre and compare distances to verify preservation.
• Coordinate practice: Work problems rotating points and figures by 90°, 180°, and 270° about the origin and about other points (use the translate-rotate-translate method).
• Discovery task: Give pupils pairs of shapes (figure & image). Let them find the centre and angle by constructing perpendicular bisectors; check results with GeoGebra.
• Symmetry hunt: Students bring photos/drawings of Kenyan objects (baskets, fabrics, wheels) and determine the order of rotational symmetry.
• 3D discussion: Using simple models (toy cube, plastic bottle), ask learners to identify axes of rotation and discuss order (rotate model and observe when it looks the same).
• Problem solving: Prove congruence by presenting a rotation that maps one triangle to another; ask learners to write the congruence statement and corresponding sides/angles.
• Assessment ideas: short test with construction tasks, coordinate rotation calculations, and identification of symmetry orders in provided shapes.

Practice exercises (with brief answers)

1. Rotate point P(3, 2) about the origin by 90° CCW. Answer: (−2, 3).
2. Rotate the triangle with vertices (1,0), (2,1), (1,2) by 180° about (0,0). Answer: (−1,0), (−2,−1), (−1,−2).
3. Given A(1,2) → A'(−2,1) and B(3,2) → B'(−2,3). Find centre O. (Construct perpendicular bisectors — answer O = (−1,0) for this example.)
4. What is the order of rotational symmetry of a regular hexagon? Answer: 6.
5. Name an axis of rotation of a cylinder and its order. Answer: axis through the centre along its length; infinite order.

Notes prepared for classroom use: include drawing practice, coordinate work and practical activities. Encourage use of GeoGebra or paper tracing for exploration. For further extension, explore compositions of rotations and connections to reflection and translation.